US 4980640 A Abstract A method of and device for the determination with high computational efficiency of spectrum parameters of a spectrum utilizes quantification and model fitting of sampling values of a signal in the time domain on the basis of a model function comprising exponentially damped sinusoids. The method starts with a coarse estimation of a number of spectrum parameters, followed by linearization of the model function and iterative estimation of the coarsely estimated parameters and further parameters, utilizing a least-squares optimization procedure where inner product terms are brought into analytic form and are preferably recursively determined.
Claims(13) 1. A method of determining spectrum parameters of a spectrum related to spectroscopic signals, in which method the spectroscopic signals derived from a substance are sampled in order to obtain sampling values and are approximated by a complex model function which contains the spectrum parameters and exponentially damped sinusoids, in which method there is also made an initial estimate, based on the sampling values, for at least one of the spectrum parameters, the at least one and further spectrum parameters being estimated from the model function by iteration with at least-squares optimisation procedure, and prior knowledge is introduced into the model function, characterized in that prior to the iteration the model function is linearised with respect to the at least one spectrum parameter, after which inner product terms occurring in the least-squares procedure are converted into an analytic form which is used during iteration.
2. A method as claimed in claim 1, characterized in that terms of the analytic form are recursively determined.
3. A method as claimed in claim 1, characterized in that the model function contains a polynomial whereby the exponentially damped sinusoids are multiplied.
4. A method as claimed in claim 1, characterized in that the initial estimate of the spectrum parameters is made by means of a non-iterative estimation procedure.
5. A method as claimed in claim 1, characterized in that the initial estimate of the spectrum parameters is made on the basis of a coarse spectrum obtained from the sampling values by Fourier transformation.
6. A method as claimed in claim 2, characterized in that the model function contains a polynomial whereby the exponentially damped sinusoids are multiplied.
7. A method as claimed in claim 2, characterized in that the initial estimate of the spectrum parameters is made by means of a non-iterative estimation procedure.
8. A method as claimed in claim 3, characterized in that the initial estimate of the spectrum parameters is made by means of a non-iterative estimation procedure.
9. A method as claimed in claim 2, characterized in that the initial estimate of the spectrum parameters is made on the basis of a coarse spectrum obtained from the sampling values by Fourier transformation.
10. A method as claimed in claim 3 characterized in that the initial estimate of the spectrum parameters is made on the basis of a coarse spectrum obtained from the sampling values by Fourier transformation.
11. A device for determining spectrum parameters of a spectrum related to spectrographic signals, which device comprises means for generating spectrographic signals in a substance, sampling means for obtaining sampling values from the generated spectrographic signals, programmed arithmetic means for determining said spectrum parameters by approximating said spectrographic signals with a complex model function which contains said spectrum parameters and exponentially damped sinusoids and is in a form introducing prior knowledge of the spectrographic signals into the model function, by making an initial estimate for at least one spectrum parameter, and by estimating based on the sampling values, using iteration, the at least one and further spectrum parameters with a least-squares optimization procedure, characterized in that prior to iteration the model function is linearized with respect to the at least one spectrum parameter, after which inner product terms occurring in the least-squares procedure are converted into an analytic form which is used during iteration.
12. A device as claimed in claim 11, characterized in that the programmed arithmetic means makes the initial estimate of the at least one spectrum parameter by a non-iterative estimation procedure.
13. A device as claimed in claim 11, wherein said programmed means further comprises means for determining a coarse spectrum by Fourier transformation of the sampling values, display means for displaying said coarse spectrum, and a cursor control device coupled to the programmed means for picking up parameters from the coarse spectrum for in making said initial estimate.
Description 1. Field of the Invention The invention relates to a method of determining spectrum parameters of a spectrum relates to spectroscopic signals, in which method the spectroscopic signals derived from a substance are sampled in order to obtain sampling values and are approximated by a complex model function which contains the spectrum parameters and exponentially damped sinusoids, in which method there is also made an initial estimate, based on the sampling values, for at least one of the spectrum parameters, the at least one and further spectrum parameters being accurately estimated from the model function by iteration with a least-squares optimisation procedure, and prior knowledge is introduced into the model function. The invention also relates to a device for determining spectrum parameters of a spectrum related to spectroscopic signals, which device comprises means for generating the spectroscopic signals in a substance, sampling means for obtaining sampling values from the spectroscopic signals, and displaying means for displaying the spectrum, and also comprises programmed means for making an initial estimate for at least one spectrum parameter, which programmed means also comprise a model function of exponentially damped sinusoids and are suitable for storing prior knowledge of the spectroscopic signals and for executing a least-square optimisation procedure on the basis of the sampling values, the programmed means furthermore comprising iteration means for accurately estimating, the at least one and further spectrum parameters by means of the least-squares optimisation procedure. 2. Description of the Prior Art A method of this kind is inter alia suitable for signals which are mainly exponentially damped, for example for signals obtained during a magnetic resonance experiment from an entire body as the substance as well as from a part of the body. The method can also be used, for example for X-ray spectroscopy or FT infrared spectroscopy. A method of this kind is disclosed in an article by J. W. C. van der Veen et al, "Accurate Quantification of in Vivo It is an object of the invention to provide a method which, when the model function contains exponentially damped sinusoids, requires substantially less computation time when executed by means of a computer. To achieve this, a method in accordance with the invention is characterized in that prior to the iteration the model function is linearised with respect to the at least one spectrum parameter, after which inner product terms occurring in the least-squares procedure are internally converted into an analytic form which is used during iteration. The number of columns of the matrix is increased with respect to the matrix in said article due to residual parameters which remain after linearisation, for example frequency and damping. The invention is based on the idea to convert first the inner product terms into analytic form in order to solve the least-squares problem in the case of exponentially damped sinusoids as the model function, rather than calculate, like in the known method, all matrix products one term after the other for all possible summations by means of row and column multiplications. The computation time is thus substantially reduced. In comparison with the known method, the method in accordance with the invention is 30 times faster in the case of 17 sinusoids and 512 sampling values. Random tests have shown that the Cramer-Rao lower limit for estimating the parameters is at least substantially reached, so that the method produces reliable parameters. The computation time increases in proportion to the first power of the number of sampling values, assuming that the other parameters, for example the number of sinusoids, remain the same. A version of a method in accordance with the invention is characterized in that terms of the analytic form are recursively determined. Thus, the method can be simply implemented in a computer. A version of a method in accordance with the invention is characterized in that the model function contains a polynomial whereby the exponentially damped sinusoids are multiplied. A sum of exponentially damped sinusoids does not always constitute a good model function in all circumstances. By multiplication by a polynomial, a model function is obtained which is adapted to circumstances. The inner product terms retain their comparatively simple form. A version of a method in accordance with the invention is characterized in that the initial estimate of the spectrum parameters is made by means of a non-iterative estimation procedure. Initial values can thus be determined for parameters which occur non-linearly in the model function. For a non-iterative estimation procedure HSVD (Hankel matrix singular value decomposition) can be used as described in "Improved Algorithm for Noniterative Time-Domain Model Fitting to Exponentially Damped Magnetic Resonance Signals", H. Barkhuijsen et al, Journal of Magnetic Resonance 73, pp. 553-557, 1987. A preferred version of a method in accordance with the invention is characterized in that the initial estimate of the spectrum parameters is made on the basis of a coarse spectrum obtained from the sampling values by Fourier transformation. In the case of a comparatively poor signal-to-noise ratio, first a filtering operation can be performed. Initial values of the parameters which occur non-linearly in the model function can be determined from the coarse spectrum. These parameters, frequency and damping, can be determined directly from a display screen on which the coarse spectrum is displayed by picking up, using a so-called mouse, at peak locations in the coarse spectrum the peak locations themselves (frequencies) and the peak width at half peak height (a measure of the damping) for supply to the programmed means in a computer. When the coarse spectrum contains excessive phase distortion, first a phase correction method incorporated in the programmed means can be executed, possibly after filtering (in the case of a comparatively poor signal-to-noise ratio), for example as described in an article by C. H. Sotak et al in Journal of Magnetic Resonance, Vol. 57, pp. 453-462, 1984. The phase correction of the coarse spectrum and the determination of the initial values can also be performed by means of a method as described in the non-prepublished Netherlands Patent Application No. 8702701 corresponding to the commonly owned U.S. patent application Ser. No. 270,923, filed Nov. 14, 1988, entitled "Method of and Device for Automatic Phase Correction of Complex NMR Spectra". The invention will be described in detail hereinafter with reference to a drawing, therein FIG. 1 diagrammatically shows a device in accordance with the invention. FIG. 2 shows, in the form of a matrix, a set of linear equations in accordance with the invention, to be used for each iteration step during the iteration process, FIG. 3 shows the analytic form to be substituted, in accordance with the invention, in the set of linear equations given in FIG. 2, FIG. 4 shows a spectrum obtained by means of the method in accordance with the invention, and FIG. 5 shows a table containing parameters of the spectrum of FIG. 4. FIG. 1 diagrammatically shows a device 1 in accordance with the invention. Specifically a magnetic resonance device for obtaining spectroscopic signals is shown by way of example. The device 1 comprises magnet coils 3 for generating a steady, uniform magnetic field B FIG. 2 shows a set of linear equations (2A) to be used in accordance with the invention, for each iteration step during the iteration process. After an initial estimate of at least one spectrum parameter, frequency and damping of peaks occurring in the spectrum in the present example, the parameters frequency and damping and also amplitude are accurately estimated in a number of iteration steps on the basis of the set (2A). Therein, the column vector c relates to the amplitudes of the peaks of the spectrum and Re c represents the residual parameters Δρ of the damping ρ and Im c represents the residual parameters Δnu of the frequency nu, M1 to M6 being matrices in the partitioned form of the extended Fisher data matrix M of the least-squares problem, extended with columns for the residual parameters Δρ and Δnu for all peaks, and v
x where x
[c+(n+δ)c'].exp[(ρ in x
f After each iteration step the current values of ρ FIG. 3 shows the analytic form for the inner product terms to be substituted in the set of linear equations shown in FIG. 2. By elaborating in the formules (2B) to (2G) the product f
z=exp[ρ Formules (3B) and (3C) show recursive relations for the respective terms corresponding to l=0, 1 and 2. When z approximates 1, an inaccuracy may occur when use is made of the recursive formules (3B) and (3C). By introducing the variable eps=z-1, this problem can be circumvented. (3D) to (3F) show formules for use in such a case. The computation time required will then slightly increase again, notably in the case of a large N, because of the combinations occurring therein, for example when N exceeds n+1. Without substantial loss of accuracy truncation can take place for a suitable value of n, so that the computation efficiency is increased again. For example, in the case of 512 sampling values it will suffice to truncate the summing operations for, for example n=10. FIG. 4 shows a spectrum obtained by means of the method in accordance with the invention for in vivo FIG. 5 shows a table containing the parameters of the spectrum 4c in FIG. 4. The number of exponentially damped sinusoids was initially set at 25; it was found that 17 sinusoids were of spectroscopic relevance. For each resonance peak there are shown parameters obtained using a method other than the method in accordance with the invention, in this case by means of HSVD without utilising prior knowledge and therebelow the parameters obtained utilising the method in accordance with the invention with introduction of prior knowledge. The errors indicated amount to twice the standard deviation. Many alternatives are feasible for those skilled in the art without departing from the scope of the invention. For example, in the case of NMR many pulse sequences can be used for spectroscopy in order to obtain resonance signals, both volume selective as well as non-volume selective. For a volume-selective pulse sequence use can be made of the SPARS method, as disclosed in Magnetic Resonance Imaging, Vol. 4, pp. 237-239, 1986. Even more prior knowledge can be used, for example information concerning multiplets in the spectrum. For a given signal-to-noise ratio for in vivo NMR, the use of prior knowledge will often be required in order to achieve suitable quantification. Patent Citations
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